# If f(x)= cos 4 x  and g(x) = -3x , how do you differentiate f(g(x))  using the chain rule?

May 12, 2016

d/dx(f(g(x))=-12sin12x

#### Explanation:

As $f \left(x\right) = \cos 4 x$ and $g \left(x\right) = - 3 x$, $f \left(g \left(x\right)\right) = \cos 4 \left(- 3 x\right)$

According to chain rule

$\frac{d}{\mathrm{dx}} f \left(g \left(x\right)\right) = \frac{\mathrm{df}}{\mathrm{dg}} \times \frac{\mathrm{dg}}{\mathrm{dx}}$

Hence, as $f \left(g \left(x\right)\right) = \cos 4 \left(- 3 x\right)$

d/dx(f(g(x))=-4sin4(-3x) xx(-3)

= $12 \sin \left(- 12 x\right) = - 12 \sin 12 x$

May 12, 2016

$- 12 \sin \left(12 x\right)$

#### Explanation:

$\frac{\mathrm{df}}{\mathrm{dx}} = \frac{\mathrm{df}}{\mathrm{dg}} \frac{\mathrm{dg}}{\mathrm{dx}}$
$= - 4 \sin \left(4 g \left(x\right)\right) \times \left(- 3\right)$
$= 12 \sin \left(4 \left(- 3 x\right)\right)$
$- 12 \sin \left(12 x\right)$